Edoburun, who was called up to the team after Reece Prescod was forced out with injury, finished clear of Trinidad and Tobago in 38.35 seconds.
Jona Efoloko, 22, was preferred to Adam Gemili on the opening leg, with Zharnel Hughes and Nethaneel Mitchell-Blake impressing in the middle of the race.
England’s women took silver behind Nigeria in the women’s version.
Led off by Asha Philip, Imani Lansiquot and Bianca Williams delivered the baton into the final 100m with ground to make up on Nigeria.
Anchor Daryll Neita briefly threatened to reel in Nzubechi Grace Nwokocha as the crowd sensed a dramatic comeback, but England ran out of track.
Nigeria won in 42.10, 0.31secs ahead of the hosts.
Efoloko, a former world under-20 200m champion, was also alongside Mitchell-Blake and Hughes when Great Britain won world bronze in Eugene last month.
“I am loving it more and more. Second time, second medal. It’s a lot of pressure and you want to do well,” he told BBC Sport.
He was waiting to embrace Edoburun at the end of the race as the pair exhorted the crowd to make even more noise.
“I am just so happy to come in and fit right in with the boys,” said Edoburun.
“I didn’t want to miss out on this special generation of athletes and I want to take this confidence into the Europeans.”